A Bayes procedure for the identification of univariate time series models
type: Item
instance of: scholarly article
title: A Bayes procedure for the identification of univariate time series models
zbMATH Open document ID: 0603.62093
author: D. S. Poskitt
published in: The Annals of Statistics
publication date: 1986-01-01T00:00:00Z
review text: The problem studied is the selection of a time series model from a given class of such models, using Bayesian arguments. A key result is that the posterior expected utility is maximized asymptotically by selecting the model which minimizes the criterion function \[ \Delta \{m({\hat \theta}\underset \tilde{} {\;}_ T)\}=2^{-1}Tl_ T({\hat \theta}\underset \tilde{} {\;}_ T)+2^{-1}d \ln T+\eta_ T\{m({\hat \theta}\underset \tilde{} {\;}_ T)\}, \] where T is the sample size, \({\hat \theta}_ T\) is a maximum likelihood estimator of the parameter \({\underset \tilde{} \theta}\in \Theta\) (so that asymptotically the likelihood function \(l_ T\) behaves like the kernel of a d variate Gaussian density with mean vector \({\hat \theta}_ T)\), \(m({\underset \tilde{} \theta})\) denotes the action of choosing \({\underset \tilde{} \theta}\in \Theta\), and \(\eta_ T\) is the integrated square relative error, \(\eta_ T=\int^{\pi}_{- \pi}[f(\omega)/g({\underset \tilde{} \theta},\omega)-1]^ 2 d\omega\), where \(f(\omega)\) is the power spectrum and \(g({\underset \tilde{} \theta},\omega)\) a particular functional parametric form. In a corollary, \(\Delta\) is specified for choosing one ARMA(p,d) model, and the general and particular criteria are related to the work of other researchers. In terms of ARMA models the author expresses that ''the criterion function leads to the selection of the model that appears to provide the best compromise between predictive characteristics, the autocorrelation structure of the residuals and model dimensionality''. In a final section consistency of the criterion is proved.
Mathematics Subject Classification ID: 62M10
Mathematics Subject Classification ID: 62C10
zbMATH DE Number: 3974122
zbMATH Keywords: identification criterion
zbMATH Keywords: order determination
zbMATH Keywords: one-step ahead prediction error variance
zbMATH Keywords: autocorrelations of the residual process
zbMATH Keywords: model selection
zbMATH Keywords: autoregressive moving-average
zbMATH Keywords: Bayes decision rule
zbMATH Keywords: time series
zbMATH Keywords: posterior expected utility
zbMATH Keywords: criterion function
zbMATH Keywords: Gaussian density
zbMATH Keywords: integrated square relative error
zbMATH Keywords: power spectrum
zbMATH Keywords: ARMA
zbMATH Keywords: consistency
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aos/1176349935
OpenAlex ID: W2048489184
DOI: 10.1214/AOS/1176349935
description: scientific article; zbMATH DE number 3974122